|
|
| |
| THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS |
| |
Citation： |
Hu Ke.THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS[J].Chinese Annals of Mathematics B,1983,4(2):187~190 |
| Page view： 678
Net amount： 685 |
Authors： |
Hu Ke; |
|
|
| Abstract： |
In this paper, the author obtains the following results
(1)If Taylor coefflients of a function $\[w(z) = \sum\limits_{n = 1}^\infty {{A_n}{z^n}} \]$ satisfy the conditions
(i)$\[\sum\limits_{k = 1}^\infty {k{{\left| {{A_k}} \right|}^2}} < \infty \]$
(ii)Re$\[\sum\limits_{k = 1}^\infty {{A_k}} = O(1)(n \to \infty )\]$
(iii)$\[{A_k} = O(\frac{1}{k})\]$, the for any h>0 the function $\[\varphi (z) = \exp \{ w(z)\} = \sum\limits_{k = 0}^\infty {{D_k}} {z^k}\]$ satisfies the asymptotic equality
$$\[\left| {\frac{{{{\{ \varphi (z){{(1 - z)}^{ - h}}\} }_n}}}{{{d_n}(h)}} - \sum\limits_{k = 0}^n {{D_k}} } \right| = o(l)(n \to \infty )\]$$,
the case $\[h > \frac{1}{2}\]$ was proved by Milin
(2)If $\[f(z) = z + {a_2}{z^2} + \cdots \in {S^*}\]$ and $\[\mathop {\lim }\limits_{r \to 1} \frac{{(1 - {r^2})}}{r}\mathop {\max }\limits_{\left| z \right| = r} \left| {f(z)} \right| = \alpha \]$, then for $\[\lambda > \frac{1}{2}\]$
$$\[\mathop {\lim }\limits_{n \to \infty } \frac{{\left| {\left| {{{\{ {{(\frac{{f(z)}}{z})}^\lambda }\} }_n}} \right| - \left| {{{\{ {{(\frac{{f(z)}}{z})}^\lambda }\} }_{n - 1}}} \right|} \right|}}{{{d_n}(2\lambda - 1)}} = {\alpha ^\lambda }\]$$ |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
|
|
|
